Finite Group Databasebeta

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ΓL(1,𝔽8)×#1157⋊C3 #72245276

order7056
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3

Aliases

direct productΓL(1,𝔽8)×#1157⋊C3

Splittings

FittingC2×C4×C72 ⋊ C2×C32
HallC7×#1157 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ #41
SylowC72 ⋊ C32×#41

Subgroups

subgroupG/H
centerC4C2×ΓL(1,𝔽8)×AGL(1,𝔽7)
commutator, socle, cosocleC2×C72C23×C32
FrattiniC2C22×ΓL(1,𝔽8)×AGL(1,𝔽7)
FittingC2×C4×C72C2×C32
hypercenterC2×C4ΓL(1,𝔽8)×AGL(1,𝔽7)
hypocenterC72C32×#41
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C4 ◁ C2×C4

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245383, ΓL(1,𝔽8)×#6495⋊C3, #72245459, …
maximal normal subgroup of#72245383, ΓL(1,𝔽8)×#6495⋊C3, #72245459, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.